Notice that the [http://mathworld.wolfram.com/Chi-SquaredDistribution.html Chi-square distribution] is not symmetric (it is positively skewed). You can visualize the Chi-Square distribution and compute all critical values either using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Chi-Square Distribution] or using the [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR Chi-square distribution calculator].

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Notice that the [http://mathworld.wolfram.com/Chi-SquaredDistribution.html Chi-Square Distribution] is not symmetric (it is positively skewed). You can visualize the Chi-Square distribution and compute all critical values either using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Chi-Square Distribution] or using the [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm SOCR Chi-Square Distribution Calculator].

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The [http://mathworld.wolfram.com/F-Distribution.html Fisher's F distribution], and the corresponding F-test, is used to test if the variances of two populations are equal. Depending on the alternative hypothesis, we can use either a two-tailed test or a one-tailed test. The two-tailed version tests against an alternative that the standard deviations are not equal (<math>H_1: \sigma_1^2 \not= \sigma_2^2</math>). The one-tailed version only tests in one direction (<math>H_1: \sigma_1^2 < \sigma_2^2</math> or <math>H_1: \sigma_1^2 > \sigma_2^2</math>). The choice is determined by the [[AP_Statistics_Curriculum_2007_IntroDesign | study design]] before any data is analyzed. For example, if a modification to an existent medical treatment is proposed, we may only be interested in knowing if the new treatment is more consistent and less variable than the established medical intervention.

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The [http://mathworld.wolfram.com/F-Distribution.html Fisher's F Distribution], and the corresponding F-test, is used to test if the variances of two populations are equal. Depending on the alternative hypothesis, we can use either a two-tailed test or a one-tailed test. The two-tailed version tests against an alternative that the standard deviations are not equal (<math>H_1: \sigma_1^2 \not= \sigma_2^2</math>). The one-tailed version only tests in one direction (<math>H_1: \sigma_1^2 < \sigma_2^2</math> or <math>H_1: \sigma_1^2 > \sigma_2^2</math>). The choice is determined by the [[AP_Statistics_Curriculum_2007_IntroDesign | study design]] before any data is analyzed. For example, if a modification to an existent medical treatment is proposed, we may only be interested in knowing if the new treatment is more consistent and less variable than the established medical intervention.

Comparing Two Variances (?)

Suppose we study two populations which are approximately Normally distributed, and we take a random sample from each population, {} and {}. Recall that and have and distributions. We are interested in assessing vs. , where s1 and σ1, and s2 and σ2 and the sample and the population standard deviations for the two populations/samples, respectively.

The Fisher's F Distribution, and the corresponding F-test, is used to test if the variances of two populations are equal. Depending on the alternative hypothesis, we can use either a two-tailed test or a one-tailed test. The two-tailed version tests against an alternative that the standard deviations are not equal (). The one-tailed version only tests in one direction ( or ). The choice is determined by the study design before any data is analyzed. For example, if a modification to an existent medical treatment is proposed, we may only be interested in knowing if the new treatment is more consistent and less variable than the established medical intervention.

Test Statistic:

The farter away this ratio is from 1, the stronger the evidence for unequal population variances.

Inference: Suppose we test at significance level α = 0.05. Then the hypothesis that the two standard deviations are equal is rejected if the test statistics is outside this interval

Comparing Two Standard Deviations (σ1 = σ2?)

Two make inference on whether the standard deviations of two populations are equal we calculate the sample variances and apply the inference on the ratio of the sample variance using the F-test, as described above.

Hands-on activities

Formulate appropriate hypotheses and assess the significance of the evidence to reject the null hypothesis that the variances of the two populations, that the following data come from, are distinct. Assume the observations below represent random samples (of sizes 6 and 10) from two Normally distributed populations of liquid content (in fluid ounces) of beverage cans. Use (α = 0.1).

Significance Inference: P-value=. This p-value does not indicate strong evidence in the data to reject the null hypothesis. That is, the data does not have power to discriminate between the population variances of the two populations based on these (small) samples.

More examples

Use the hot-dogs dataset to formulate and test hypotheses about the difference of the population standard deviations of sodium between the poultry and the meet based hot-dogs. Repeat this with variances of calories between the beef and meet based hot-dogs.